Talk:Module (mathematics)

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Contents 1 Two questions 2 ?? 3 bimodules over commutative rings 4 Module (category theory) 5 Modules are not modular 6 faithful module 7 lead section 8 fomal should be formal

[edit] Two questions

1) In the fifth example, it is written

the category of C∞(X)-modules and the category of vector bundles over X are equivalent.

Is this actually true? I think it should be the category of finite-dimensional projective C-infinity modules, based on the Swan's theorem page.

2) What is a pseudo-module, in the parlance of Bourbaki? This is what I glanced at the article to look up in the first place. (I tried a google search to no avail. Comp sci usage and other mathematical usage overwhelms whatever meaning might be there.)

Thanks. 128.135.100.161 07:29, 1 December 2006 (UTC)

Bourbaki takes all rings to have identity and the identity to act trivially on all modules. A pseudo-module over a pseudo-ring (ring without identity) is what is called a module here. Changing conventions can be somewhat annoying... 128.135.100.161 01:02, 3 March 2007 (UTC)

[edit] ??

All ring modules are not unital. --S. A. G.

I think above refers to a problem with the article, that I think is fixed ?? linas 00:38, 6 Apr 2005 (UTC)

I agree, and the glossary of ring theory does not even stipulate (which is a good thing!) that a ring must be unital (i.e. a unit ring). But this is somehow an eternal debate... — MFH: Talk 21:03, 12 May 2005 (UTC)

[edit] bimodules over commutative rings

I'm not so sure if the statement

If R is commutative, then left R-modules are the same as
right R-modules and are simply called R-modules.

is correct, or at least, not misleading. I think one can consider a bimodule where the left and right action of the ring are completely different operations. (I think people working on Hopf groups, e.g. in noncommutative geometry, consider such things, but there seems to be no info on this here.) — MFH: Talk 21:03, 12 May 2005 (UTC)

[edit] Module (category theory)

Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 00:15, 29 Dec 2004 (UTC)

Maybe the Monad (category theory) people can help.--80.133.106.228 14:24, 28 Feb 2005 (UTC)

Maybe Module (model theory) makes more sense. linas 16:09, 12 Mar 2005 (UTC)

[edit] Modules are not modular

The redirection Modular -> Module -> Module (mathematics) is nonsense because there is no module corresponding to modular forms (and related concepts), nor in modular representation theory.--80.133.106.228 14:18, 28 Feb 2005 (UTC)

I fixed the disambig links for the above complaint. linas 17:14, 12 Mar 2005 (UTC)

[edit] faithful module

"Faithful. A faithful module M is one where the action of each r in R gives an injective map M→M. Equivalently, the annihilator of M is the zero ideal." This is incorrect, a faithful module is one where the action of each nonzero r in R is not the trivial map, i.e. if a does not equal b in R, then a->ax and b->bx are different endomorphisms. These statements are equivalent to ann(M)=0.

[edit] lead section

I think this article (like all articles) needs a good lead section, and I think the current

In abstract algebra, the notion of a module over a ring is the common generalization of two of the most important notions in algebra, vector space, and abelian group.

is doing a bad job. It doesn't actually define what a module is, the abstract algebra classification makes modules sound vastly less important than they actually are (it sounds like it's some exotic structure of interest mostly in universal algebra, like a magma actually is (apologies to anyone deeply in love with magmas, but they do seem exotic right now)).

I do not see how module is the common generalisation of vector spaces (F-modules, where F is a field) and abelian groups (Z-modules); both of those are actually, in modern terminology, R-R-bimodules, and that would be their common generalisation: R-R-bimodules where R is a commutative domain. Going all the way to real modules is quite a bit more general; noncommutative rings, in particular, are important.

(I'm also curious about vector spaces being such an important notion - instructive, maybe, but nowadays you just think of them as modules over a field.)

I'd suggest something along the lines of:

In mathematics, a module over a ring is a space whose elements can be added and multiplied by elements of the ring, analogously to vector spaces. Formally, a module M over R is an abelian group with a ring homomorphism R → Hom(M, M).

But that doesn't sound quite good enough for me to be bold without asking for better proposals first.

RandomP 19:47, 29 April 2006 (UTC)

I like the current intro more however, it reads better, is more elementary and more motivational than what you suggest I think. Maybe you can add the sentences you want under the current intro rather than replacing it? Oleg Alexandrov (talk) 20:23, 29 April 2006 (UTC)

The audience of this article includes people with only the barest knowledge of abstract algebra, and I think for these people the current introduction provides more motivation. There's nothing stopping the more formal, technical definition from appearing lower down. Certainly there should be no mention of Hom in the introduction, it's unnecessarily technical. On the other hand, I do agree that the statement that modules are a common generalisation of vector spaces and abelian groups is a bit misleading. I noticed this some time ago and wasn't quite sure how to fix it. Perhaps it's fair to say that "module" is a generalisation of "vector space", but not so fair to call it a generalisation of "abelian group". That would after all be much the same as saying that a vector space is a generalisation of an abelian group, which sounds a bit silly. Rather I would say that a module is an abelian group with additional structure. Dmharvey 20:59, 29 April 2006 (UTC)

Well, an abelian group is simply a module over Z. So, it is true that a module is a more general case of a group, and also of a vector space. No? Oleg Alexandrov (talk) 21:04, 29 April 2006 (UTC)

Yes, but it is by no means the simplest way to generalise both. Furthermore, abelian groups are rarely thought of as having an external multiplication with Z until after they're proved to be Z-modules. I'd suggest saying they also generalise vector spaces over skew fields, but those are introduced rarely, I fear.

In any case, the blatant bias towards the commutative case needs to go!

RandomP 21:28, 29 April 2006 (UTC)

Edit conflict: We were talking about this issue at talk:manifold last week. Dmharvey is right in general, it doesn't have much meaning to call something with extra structure a generalization of something with less structure. In general, it really only makes sense to talk about things which have fewer axioms (but the same size structure) a generalization. Thus, monoids are generalizations of groups, topological manifolds are generalizations of smooth manifolds, and Banach spaces are generalizations of Euclidean space, but rings are not generalizations of groups (rings have more structure, not every group can be a ring), manifolds are not generalizations of topological spaces (not every space can be a manifold), and Banach algebras are not generalizations of Euclidean space. But Oleg is also right: in this instance, every abelian group is a module. The extra structure that modules have is somehow degenerate in the case of Z-modules. I wonder what the technical explanation is for what's going on there. Nevertheless, for arbitrary modules, that extra structure is nontrivial, and therefore it may not make much sense to call a module a generalization of an abelian group. Even if it does make sense, is it useful in practice? Does it merit mention in the intro? -lethe ^{talk +} 21:29, 29 April 2006 (UTC)

I guess this "degeneracy" that I'm wondering about stems from the fact that Z is a free ring, so the functor from R to Ab is determined by its action on the underlying object, the terminal category, which is just a choice of an abelian group. -lethe ^{talk +} 21:49, 29 April 2006 (UTC)

Instead of this phrase "common generalization", why not something less convoluted, like "a module generalizes both vector spaces and abelian groups"? -lethe ^{talk +} 21:17, 29 April 2006 (UTC)

I'd definitely not want to say that "a module generalizes ... abelian groups", for the same reason Dmharvey stated. This is very misleading. It suffices to note that abelian groups "are the same things as" Z-modules. — merge 10:01, 30 April 2006 (UTC)

I like Dmharvey's new edit. -lethe ^{talk +} 13:44, 30 April 2006 (UTC)

It's definitely an improvement! However, now it suggests, rather misleadingly, that modules are important only for commutative algebra, which itself is important in a number of other fields. The noncommutative case is important, and ignoring it before even given the formal definition is more misleading than skipping it alltogether.

What modules are useful for might change, but what modules are is not going to.

RandomP 17:54, 30 April 2006 (UTC)

Unfortunately I am quite ignorant when it comes to modules over non-commutative rings. The only such objects I ever deal with would be modules over a group ring such as Z[G], so I guess this is really just representation theory, which is mentioned in the introduction. If you can suggest other areas where modules over non-commutative rings come up, you're more than welcome to add them to the intro. Dmharvey 18:04, 30 April 2006 (UTC)

The point is that the representation theory of groups, for example, needs modules, not commutative algebra.

Modules are one of the core notions of commutative algebra, which is essential in many important fields of mathematics, including algebraic geometry, homological algebra, algebraic topology, and the representation theory of groups.

suggests that all modules are good for is commutative algebra.

RandomP 18:57, 30 April 2006 (UTC)

FWIW, I'm not usually being bold with the lead section precisely because others might prefer the way it currently stands. Now there appears to be some consensus that we need a new lead section, that's maybe not so much of an issue.

I still think that the important thing, and the thing the reader should actually gain from the article, is the definition. Yes, modules are important for pretty much everything even vaguely research-related (in pure mathematics and non-classical theoretical physics, at least) today, but really, knowing what a module is is more important than knowing what it's good for.

Currently, using a half-screen browser layout, the definition (and let's be clear here: the "formal" in there isn't really required) just barely still fits on the same page. A linear reader will stumble over all the complicated scary terms, and quite possibly give up on the article, before even being told what a module is.

I'm not the ideal person to fix that; I'm just not a terribly good writer. But I can point out that it's wrong, and that I'd consider even the clumsy

A module M over a ring R is an abelian group (M, +) together with a multiplication operation R × M → M.

better: it's a complete definition, since the term "multiplication" implies distributivity and associativity.

I can live with a non-scary sentence about that really being the same thing as a vector space preceding that; the current best practice appears to be that fields and vector spaces come first, then rings and modules, so no reader who's scared by vector spaces is going to profit much from learning about modules (I don't consider this a general problem, it's more of a deficiency in the literature). But the definition is the important bit, and if there's no good way to say it without using symbols, well, we'll just have to use symbols instead.

RandomP 20:18, 30 April 2006 (UTC)

Agree that "formal" is not required.

Disagree that you're not a good writer. You seem eminently capable of stringing words together.

Agree that the "definition" section is too long. Some of that material should be put in a separate section.

The trick here is to be serving two audiences at the same time. On the one hand we want the concise version "an R-module is a ring homomorphism from R to End M for an abelian group M", and on the other hand we want something accessible to someone learning this kind of algebra for the first time. I'm actually thinking in this case that it might be okay to put the "advanced" version first, as long as it's very brief, and then immediately following that have something like "Explicitly, this means the following:...", and spell out the "multiplication map", with the properties that it needs to have, as we have listed currently. Usually, I would insist on the baby version up front, but here it seems like it might not be such a problem. Dmharvey 20:41, 30 April 2006 (UTC)

Here's an attempt with a symbol-free version, with the baby version kept as "formal definition". I really think this might be the main problem; there just isn't a good English-language description of modules.

what a module really is, I'd say, is a thingy that a ring acts on. Of course that kind of requires it to have an addition, and thus be an abelian group, but you don't really think of it that way. Or do you?

RandomP 21:07, 30 April 2006 (UTC)

Perhaps modules generalize ideals? 128.135.100.161 01:06, 3 March 2007 (UTC)

[edit] fomal should be formal

fomal should be formal

Thanks for pointing that out! It might have saved you some time to be bold and edit it yourself, but pointing out problems on the talk page is perfectly fine, too, of course.

RandomP 21:21, 14 May 2006 (UTC)

I see now, to me it looked like a tautology, I missed the fac that an 'r' was missing in the first "formal". Oleg Alexandrov (talk) 01:20, 15 May 2006 (UTC)

FAC's have missing r's? linas 23:40, 11 March 2007 (UTC)